Locally supported, quasi-interpolatory bases for the approximation of functions on graphs
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2021
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866910367351832576 |
|---|---|
| author | Fuselier, Edward J. Ward, John Paul |
| author_facet | Fuselier, Edward J. Ward, John Paul |
| contents | Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present $\ell_\infty$ error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2101_02256 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Locally supported, quasi-interpolatory bases for the approximation of functions on graphs Fuselier, Edward J. Ward, John Paul Numerical Analysis Classical Analysis and ODEs 05C90, 41A05, 41A15, 65D07 Graph-based approximation methods are of growing interest in many areas, including transportation, biological and chemical networks, financial models, image processing, network flows, and more. In these applications, often a basis for the approximation space is not available analytically and must be computed. We propose perturbations of Lagrange bases on graphs, where the Lagrange functions come from a class of functions analogous to classical splines. The basis functions we consider have local support, with each basis function obtained by solving a small energy minimization problem related to a differential operator on the graph. We present $\ell_\infty$ error estimates between the local basis and the corresponding interpolatory Lagrange basis functions in cases where the underlying graph satisfies a mild assumption on the connections of vertices where the function is not known, and the theoretical bounds are examined further in numerical experiments. Included in our analysis is a mixed-norm inequality for positive definite matrices that is tighter than the general estimate $\|A\|_{\infty} \leq \sqrt{n} \|A\|_{2}$. |
| title | Locally supported, quasi-interpolatory bases for the approximation of functions on graphs |
| topic | Numerical Analysis Classical Analysis and ODEs 05C90, 41A05, 41A15, 65D07 |
| url | https://arxiv.org/abs/2101.02256 |